| L(s) = 1 | + (−0.184 − 2.58i)2-s + (4.62 + 1.00i)3-s + (−2.68 + 0.386i)4-s + (0.815 + 4.93i)5-s + (1.74 − 12.1i)6-s + (0.277 − 0.508i)7-s + (−0.707 − 3.25i)8-s + (12.1 + 5.56i)9-s + (12.6 − 3.01i)10-s + (1.72 + 1.99i)11-s + (−12.8 − 0.916i)12-s + (−3.62 − 6.62i)13-s + (−1.36 − 0.623i)14-s + (−1.19 + 23.6i)15-s + (−18.6 + 5.49i)16-s + (−1.02 − 1.37i)17-s + ⋯ |
| L(s) = 1 | + (−0.0924 − 1.29i)2-s + (1.54 + 0.335i)3-s + (−0.671 + 0.0965i)4-s + (0.163 + 0.986i)5-s + (0.290 − 2.02i)6-s + (0.0396 − 0.0726i)7-s + (−0.0884 − 0.406i)8-s + (1.35 + 0.618i)9-s + (1.26 − 0.301i)10-s + (0.157 + 0.181i)11-s + (−1.06 − 0.0763i)12-s + (−0.278 − 0.509i)13-s + (−0.0975 − 0.0445i)14-s + (−0.0795 + 1.57i)15-s + (−1.16 + 0.343i)16-s + (−0.0604 − 0.0807i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.80004 - 1.04036i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.80004 - 1.04036i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.815 - 4.93i)T \) |
| 23 | \( 1 + (4.80 + 22.4i)T \) |
| good | 2 | \( 1 + (0.184 + 2.58i)T + (-3.95 + 0.569i)T^{2} \) |
| 3 | \( 1 + (-4.62 - 1.00i)T + (8.18 + 3.73i)T^{2} \) |
| 7 | \( 1 + (-0.277 + 0.508i)T + (-26.4 - 41.2i)T^{2} \) |
| 11 | \( 1 + (-1.72 - 1.99i)T + (-17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (3.62 + 6.62i)T + (-91.3 + 142. i)T^{2} \) |
| 17 | \( 1 + (1.02 + 1.37i)T + (-81.4 + 277. i)T^{2} \) |
| 19 | \( 1 + (18.5 - 2.67i)T + (346. - 101. i)T^{2} \) |
| 29 | \( 1 + (33.7 + 4.85i)T + (806. + 236. i)T^{2} \) |
| 31 | \( 1 + (-39.4 - 25.3i)T + (399. + 874. i)T^{2} \) |
| 37 | \( 1 + (-23.1 - 62.0i)T + (-1.03e3 + 896. i)T^{2} \) |
| 41 | \( 1 + (30.3 + 66.4i)T + (-1.10e3 + 1.27e3i)T^{2} \) |
| 43 | \( 1 + (-31.0 - 6.74i)T + (1.68e3 + 768. i)T^{2} \) |
| 47 | \( 1 + (13.5 - 13.5i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-51.5 - 28.1i)T + (1.51e3 + 2.36e3i)T^{2} \) |
| 59 | \( 1 + (-22.3 + 76.0i)T + (-2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (18.6 + 11.9i)T + (1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (2.91 + 40.7i)T + (-4.44e3 + 638. i)T^{2} \) |
| 71 | \( 1 + (3.05 - 3.52i)T + (-717. - 4.98e3i)T^{2} \) |
| 73 | \( 1 + (-23.5 + 31.5i)T + (-1.50e3 - 5.11e3i)T^{2} \) |
| 79 | \( 1 + (15.8 - 53.9i)T + (-5.25e3 - 3.37e3i)T^{2} \) |
| 83 | \( 1 + (-17.7 + 6.60i)T + (5.20e3 - 4.51e3i)T^{2} \) |
| 89 | \( 1 + (13.3 + 20.7i)T + (-3.29e3 + 7.20e3i)T^{2} \) |
| 97 | \( 1 + (-106. - 39.5i)T + (7.11e3 + 6.16e3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.19889407537199032162534762503, −12.08478953289933495956546543499, −10.73005966071427555579833755857, −10.14332807729907722334991945694, −9.207635082758642768030671491118, −8.028555944633038796149664536874, −6.66792898632520902331351860248, −4.15275264597660558517402128284, −3.04469900131317903116628563230, −2.17353277757817623496157109685,
2.15681800450674662442189927739, 4.21042482662782255186598841757, 5.78433433338129196420254286179, 7.17733627603116600075561266075, 8.106258015771607101624361884518, 8.813743840733353111084814389813, 9.584365317557933223804617621796, 11.65307591311186614496353538331, 13.06785430593427949600596171264, 13.67315948920712205727286891962
